Viscoelastic bandgap in multilayers of inorganic–organic nanolayer interfaces

Incorporating molecular nanolayers (MNLs) at inorganic interfaces offers promise for reaping unusual enhancements in fracture energy, thermal and electrical transport. Here, we reveal that multilayering MNL-bonded inorganic interfaces can result in viscoelastic damping bandgaps. Molecular dynamics simulations of Au/octanedithiol MNL/Au multilayers reveal high-damping-loss frequency bands at 33 ≤ ν ≤ 77 GHz and 278 ≤ ν ≤ 833 GHz separated by a low-loss bandgap 77 ≤ ν ≤ 278 GHz region. The viscoelastic bandgap scales with the Au/MNL interface bonding strength and density, and MNL coverage. These results and the analyses of interfacial vibrations indicate that the viscoelastic bandgap is an interface effect that cannot be explained by weighted averages of bulk responses. These findings prognosticate a variety of possibilities for accessing and tuning novel dynamic mechanical responses in materials systems and devices with significant inorganic–organic interface fractions for many applications, e.g., smart composites and sensors with self-healing/-destructing mechanical responses.

Inside the viscoelastic bandgap, the low loss modulus G" Au/octanedithiol MNL/Au ~ 0.5 GPa, and the storage modulus G' Au/octanedithiol MNL/Au ~ 7.5 GPa; these lie within the range of values of bulk Au and methylene phases (Fig. 1b). Individual bulk crystals of Au exhibit elastic behavior G Au ~ G' Au ~ 51 GPa with minimal losses of G" Au ≤ 0.25 GPa. Methylene chain crystals exhibit viscoelastic losses between 0.7 ≤ G" MNL ≤ 8.2 GPa for the entire frequency range studied (Fig. 1b). The small peaks in the 40° ≤ δ ≤ 45° range, at ~ 357, ~ 556, and ~ 769 GHz (Fig. 1a) indicate temporal lags in the molecular assemblies in responding to applied stresses. The maximum loss G" Au/octanedithiol MNL/Au ~ 104 GPa occurs at ν max ~ 625 GHz, where G' Au/octanedithiol MNL/Au exhibits frequency-dependent signatures typical of mechanical damping processes (Fig. 1b).
The damping energy losses were, however, not only unlike that expected for bulk Au or methylene chains, but also entirely different from estimates obtained using series and parallel rules of mixtures (ROM) 10 . The parallel model defines the lower bound shear modulus magnitude G LowerROM ~ 15 to 20 GPa for 33-1429 GHz, while the series model yields the upper bound G UpperROM ~ 30 to 33 GPa. Our simulations show that G Au/octanedithiol MNL/Au is either two-to-three-fold lower than G LowerROM , or four-to-nine-fold higher than G UpperROM . This result indicates that damping in the Au/octanedithiol MNL/Au multilayer is predominantly governed by interface effects, and cannot be explained by weighted averages of bulk responses.
Interfacial bond density. The low-damping viscoelastic bandgap magnitude Δν and the band edge positions ν low and ν high are sensitive to the Au-S interfacial bond density X IBD (Fig. 2a). Increasing X IBD correlates with linear increases in the bandgap Δν = ν low − ν high and the high-damping band edge frequencies (see Fig. 2b,c). Furthermore, increasing X IBD correlates with an increase in the low-frequency band damping magnitude and a decrease in the high-frequency band width. A consequence of these behaviors is that the frequency-averaged loss modulus G'' decreases linearly with X IBD from ~ 16 GPa to ~ 11.6 GPa for 0.2 ≤ X IBD ≤ 1 (see Fig. 2d). Essentially, identical results (not shown) are obtained when we vary the bond strength connoted by X IBS . In our simulations, varying bond strength does not involve bond elimination, as is the case when altering bond density.
Interfacial molecular coverage. Decreasing the interfacial molecular coverage X IMC by removing entire molecules resulted in a greater effect (see Fig. 3) on the viscoelastic bandgap than that obtained by deactivating interfacial bonds (i.e., by decreasing X IBS and X IBD ). For example, the viscoelastic bandgap completely disappears upon decreasing X IMC by ~ 10%. This result is consistent with 11,12 the higher average energy dissipation per unit coverage expected upon decreasing molecular coverage. These findings indicate that both the molecular packing, molecular order, and interfacial bond strength strongly influence viscoelastic bandgap magnitude and position. These results point to new vistas for creating a rich variety of materials with tailored frequency-dependent viscoelastic properties.
Vibrational density of states. Calculations of the vibrational density of states (VDOS) of the Au-block center-of-mass relative to the Au/MNL interface reveals that the viscoelastic deformation to be accommodated exclusively by the interface. The out-of-plane interfacial vibration peak position shifts monotonically to higher frequencies from ~ 182 to ~ 354 GHz, with increasing interfacial strength denoted by X IBD (see Fig. 4a,b). The two The gap between in-and out-of-plane interfacial mode peaks (Fig. 4a) in vibrational density of states (VDOS) shows similar sensitivity to the bond density (Fig. 4b) as the viscoelastic gap plotted in Fig. 2c. The fact they there are not quantitively the same is expected as the VDOS is evaluated for a specific subset of interfacial motion associated with relative displacement of the whole Au and MNL slabs. Strong correlations between the viscoelastic gap and the vibrational gap provide another and powerful evidence for interfaces being centrally responsible for the remarkable mechanical responses of the organic-inorganic nanolayers revealed in this work. Spatial power loss. In order to obtain insights into interface-induced damping, we spatially mapped energy dissipation using a Berendsen thermostat and averaging the power loss rate over atomic planes across the Au/ octanedithiol MNL/Au structures. Our results (see Fig. 5) indicate 2 to 3.5-fold higher power losses in the MNLs than in the Au nanolayers, with a large interfacial power loss gradient. Higher loss in the MNL is consistent with the availability of multiple energy dissipation modes in MNLs, e.g., chain kinking, bending and rotation 13 . No such losses are observed in methylene crystal ensembles, indicating that energy loss occurs primarily in the organic MNL, but only when strongly bonded to the inorganic nanolayers.

Conclusion
In conclusion, our study has unearthed the existence of viscoelastic damping bandgaps in Au/octanedithiol MNL/Au multilayers subject to cyclic shear. The bandgap features are sensitive to, and tunable with, interfacial bond strength and density, and molecular coverage. The viscoelastic bandgap scales with the Au/MNL interfacial bonding strength and density, and MNL coverage. Our results provide compelling evidence that this behavior is an intrinsically interfacial effect that cannot be explained in terms of weighted averages of bulk responses. These findings presage a variety of possibilities for accessing and tuning novel dynamic mechanical responses in materials systems and devices with significant inorganic-organic interface fractions for applications.

Methods
Model creation. We created a unit-cell consisting of four octanedithiol molecules with the sulfur head group adsorbed on the face centered cubic hollow sites of ( √ 3 × √ 3 ) R30° surface of Au(111) using methods detailed elsewhere 15,16 . The unit-cell was repeated to 6 × 5 × 2 along x-, y-, and z-axes using the Moltemplate builder 14 to create a 5.18 × 4.98 × 5.32 nm 3 cell with Au/octanedithiol MNL/Au multilayers (see Fig. 1a). Each multilayer consists of ~ 1.2-nm-thick single-crystal Au(111) nanolayers bonded with ~ 1-nm-thick octanedithiol MD protocol. We carried out the molecular dynamics (MD) calculations using the LAMMPS package 18 .
Firstly, we equilibrated the model structures for 400 picoseconds at 300 K at zero pressure using an isothermalisobaric ensemble (NPT) and applied GHz-frequency shear strain γ = γ 0 sin2πνt on a canonical ensemble (NVT) for 1000 cycles using a method described elsewhere 8,19 . The oscillatory shear resulted in a strain amplitude γ 0 = 1% for 33 ≤ ν ≤ 1429 GHz. Our simulations used a 2-fs integration time step and a Berendsen thermostat with a 5 ps damping timescale, with periodic boundary conditions applied in all the directions. Series and parallel rules of mixtures were used for computing the upper and lower bound shear modulii of the Au/octanedithiol MNL/Au multilayers, i.e., G UpperROM = V Au G Au + V MNL G MNL and G LowerROM = G Au G MNL / (V Au G Au + V MNL G MNL ) respectively.
We obtained the vibrational density of states (VDOS) of the relative motion of Au nanolayers about the Au/ MNL interfaces by Fourier transforming the velocity autocorrelation function determined from the Au center of mass velocities monitored for 500 pico-second during equilibrium in our microcanonical ensemble (NVE) simulations.

Data availability
All the data generated or analyzed during the current study are available from the corresponding author upon reasonable request.